Problem: Given $ \overrightarrow{PQ}\perp\overrightarrow{PS}$, $ m \angle QPR = 7x - 98$, and $ m \angle RPS = 9x - 116$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Solution: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since we are given that $\overrightarrow{PQ}\perp\overrightarrow{PS}$ , we know ${m\angle QPS = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 98} + {9x - 116} = {90}$ Combine like terms: $ 16x - 214 = 90$ Add $214$ to both sides: $ 16x = 304$ Divide both sides by $16$ to find $x$ $ x = 19$ Substitute $19$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 9({19}) - 116$ Simplify: $ {m\angle RPS = 171 - 116}$ So ${m\angle RPS = 55}$.